1. Data Generation

1-1. First cell type

p <- 10
first_n <- 150
corpula_mean_first <- c(10, 0, 10, 0, 10,
                        0, 10, 0, 10, 0) # mu
cov_mat_first <- matrix(0, p, p); 
diag(cov_mat_first) <- c(20, 5, 20, 5, 20,
                         5, 20, 5, 20, 5)
norm_first <- MASS::mvrnorm(first_n, corpula_mean_first, cov_mat_first)
corpula_first <- norm_first
# nonparanormal transform
corpula_first <- apply(corpula_first, 2, function(x) {0.4 * sign(x) * abs(x)^1.6})
# this is the gaussian data we need to make nonparanormals
pairs(norm_first, asp = T, pch = 16,  col = rgb(0.5,0.5,0.5,0.5), lower.panel = NULL)

# this is the nonparanormal 
pairs(corpula_first, asp = T, pch = 16,  col = rgb(0.5,0.5,0.5,0.5), lower.panel = NULL)

1-2. Second cell type

second_n <- 200
corpula_mean_second <- c(0, 10, 0, 10, 0,
                         10, 0, 10, 0, 10) # mu
cov_mat_second <- matrix(0, p, p)
diag(cov_mat_second) <- c(5, 20, 5, 20, 5,
                          20, 5, 20, 5, 20) # Sigma
norm_second <- MASS::mvrnorm(second_n, corpula_mean_second, cov_mat_second)
corpula_second <- norm_second
# nonparanormal transform
corpula_second <- apply(corpula_second, 2, function(x) {0.4 * sign(x) * abs(x)^1.6})
# this is the gaussian data we need to make nonparanormals
pairs(norm_second, asp = T, pch = 16,  col = rgb(0.5,0.5,0.5,0.5), lower.panel = NULL)

# this is the nonparanormal 
pairs(corpula_second, asp = T, pch = 16,  col = rgb(0.5,0.5,0.5,0.5), lower.panel = NULL)

1-3. Third cell type

third_n <- 150
corpula_mean_third <- c(10, 0, 0, 10, 10, 
                        8, 8, 10, 10, 8)
cov_mat_third <- matrix(4, p, p);
diag(cov_mat_third) <- c(5, 10, 5, 10, 5, 10, 5, 10, 5, 10) # Sigma

norm_third <- MASS::mvrnorm(third_n, corpula_mean_third, cov_mat_third)
corpula_third <- norm_third
# nonparanormal transform
corpula_third <- apply(corpula_third, 2, function(x) {0.6 * sign(x) * abs(x)^1.4})
norm_expr <- MASS::mvrnorm(third_n, corpula_mean_third, cov_mat_third)
# this is the gaussian data we need to make nonparanormals
pairs(norm_third, asp = T, pch = 16,  col = rgb(0.5,0.5,0.5,0.5), lower.panel = NULL)

# this is the nonparanormal 
pairs(corpula_third, asp = T, pch = 16,  col = rgb(0.5,0.5,0.5,0.5), lower.panel = NULL)

1-4. Combine the three cell types

full_dat <- rbind(cbind(corpula_first, rep(1, first_n)), 
                 cbind(corpula_second, rep(2, second_n)),
                 cbind(corpula_third, rep(3, third_n)))

gen_dat <- full_dat[, 1:10]
gen_label <- full_dat[, 11]

pairs(gen_dat, asp = T, pch = 16,  col = gen_label, lower.panel = NULL)

pairs(gen_dat, asp = T, pch = 16,  col = rgb(0.5, 0.5, 0.5, 0.5), lower.panel = NULL)

library(reshape2) # melt function
library(ggplot2) # ggplot function
library(pcaPP) # Fast Kendall function
library(energy) # Distance Correlation
library(Hmisc) # Hoeffding's D measure
## Loading required package: lattice
## Loading required package: survival
## Loading required package: Formula
## 
## Attaching package: 'Hmisc'
## The following objects are masked from 'package:base':
## 
##     format.pval, units
library(entropy) # Mutual Information
library(minerva) # Maximum Information Coefficient
library(XICOR) # Chatterjee's Coefficient
library(dHSIC) # Hilbert Schmidt Independence Criterion
library(VineCopula) # Blomqvist's Beta

make_cormat <- function(dat_mat){
  matrix_dat <- matrix(nrow = ncol(dat_mat), ncol = ncol(dat_mat))
  cor_mat_list <- list()
  
  basic_cor <- c("pearson", "spearman")
  # find each of the correlation matrices with Pearson, Spearman, Kendall Correlation Coefficients
  for (i in 1:2){
    cor_mat <- stats::cor(dat_mat, method = basic_cor[i])
    cor_mat[upper.tri(cor_mat, diag = T)] <- NA
    cor_mat_list[[i]] <- cor_mat
  }
  
  # functions that take matrix or data.frame as input
  no_loop_function <- c(pcaPP::cor.fk, Hmisc::hoeffd,
                        minerva::mine, VineCopula::BetaMatrix)
  for (i in 3:6){
    fun <- no_loop_function[[i-2]]
    cor_mat <- fun(dat_mat)
    
    if (i == 4){ # Hoeffding's D
      cor_mat <- cor_mat$D
    } else if (i == 5){ # MIC
      cor_mat <- cor_mat$MIC
    }
    
    cor_mat[upper.tri(cor_mat, diag = T)] <- NA
    cor_mat_list[[i]] <- cor_mat
  }
  
  # functions that take two variables as input to calculate correlations.
  need_loop <- c(energy::dcor2d, entropy::discretize2d,
                 XICOR::calculateXI, dHSIC::dhsic)

  for (i in 7:10){
    fun <- need_loop[[i-6]]
    
    cor_mat <- matrix(nrow = ncol(dat_mat),
                      ncol = ncol(dat_mat))
    
    for (j in 2:ncol(dat_mat)){
      for (k in 1:(j-1)){
        
        if (i == 8){ # Mutual Information
          cor_mat[j, k] <- mi.empirical(fun(as.matrix(dat_mat[, j]),
                                            as.matrix(dat_mat[, k]),
                                            numBins1 = 20,
                                            numBins2 = 20))
        } else if (i == 10){ # HSIC
          cor_mat[j, k] <- fun(as.numeric(dat_mat[, j]),
                               as.numeric(dat_mat[, k]))$dHSIC
        } else {
          cor_mat[j, k] <- fun(as.numeric(dat_mat[, j]),
                               as.numeric(dat_mat[, k]))
        }
      }
    }
    cor_mat[upper.tri(cor_mat, diag = T)] <- NA
    cor_mat_list[[i]] <- cor_mat
  }
  
  return(cor_mat_list)
}
draw_heatmap <- function(cor_mat){
    len <- 8
    melted_cormat <- melt(cor_mat)
    melted_cormat <- melted_cormat[!is.na(melted_cormat$value),]
    break_vec <- round(as.numeric(quantile(melted_cormat$value,
                                           probs = seq(0, 1, length.out = len),
                                           na.rm = T)),
                       3)
    break_vec[1] <- break_vec[1]-1
    break_vec[len] <- break_vec[len]+1
    melted_cormat$value <- cut(melted_cormat$value, breaks = break_vec)
    heatmap_color <- unique(melted_cormat$value)
  
    heatmap <- ggplot(data = melted_cormat, aes(x = Var2, y = Var1, fill = value))+
      geom_tile() +
      ggplot2::scale_fill_manual(breaks = sort(heatmap_color), 
                                 values = rev(scales::viridis_pal(begin = 0, end = 1)
                                              (length(heatmap_color))))
    return (heatmap)
}

make_cor_heatmap <- function(dat_mat){
  cor_heatmap_list <- list()
  cor_abs_heatmap_list <- list()
  cor_mat_list <- make_cormat(dat_mat)
  for (i in 1:10){
    cor_mat <- cor_mat_list[[i]]
    cor_abs_mat <- abs(cor_mat_list[[i]])
    cor_heatmap <- draw_heatmap(cor_mat)
    cor_abs_heatmap <- draw_heatmap(cor_abs_mat)
    cor_heatmap_list[[length(cor_heatmap_list) + 1]] <- cor_heatmap
    cor_abs_heatmap_list[[length(cor_abs_heatmap_list) + 1]] <- cor_abs_heatmap
    }
  ans <- list(cor_heatmap_list, cor_abs_heatmap_list)
  return (ans)
}
lst <- make_cor_heatmap(gen_dat)
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cormat_list <- make_cormat(gen_dat)
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# lst[[1]]
lst[[1]][[4]]

cor_pearson_mat <- cormat_list[[1]]; cor_spearman_mat <- cormat_list[[2]];
cor_kendall_mat <- cormat_list[[3]]; cor_hoeffd_mat <- cormat_list[[4]];
cor_MIC_mat <- cormat_list[[5]]; cor_blomqvist_mat <- cormat_list[[6]];
cor_dist_mat <- cormat_list[[7]]; cor_MI_mat <- cormat_list[[8]];
cor_XI_mat <- cormat_list[[9]]; cor_HSIC_mat <- cormat_list[[10]];

3. Find contrast characteristics among the correlation coefficients above

3-1-1. low pearson (<0.50) and high spearman (>0.5) (linearity vs monotone)

cor_contrast1 <- (abs(cor_pearson_mat) < 0.5) & (abs(cor_spearman_mat) > 0.5)
cor_contrast_ind1 <- which(cor_contrast1, arr.ind = T)
nrow(cor_contrast_ind1)
## [1] 7

3-1-2. Visualization of low pearson (<0.50) and high spearman (>0.5) (linearity vs monotone)

par(mfrow = c(2, 5))
for (i in 1:nrow(cor_contrast_ind1)){
   index1 <- cor_contrast_ind1[i, 1]; index2 <- cor_contrast_ind1[i, 2]
   plot(gen_dat[,index1], gen_dat[,index2], col = gen_label, asp = T,
      pch = 16, xlab = paste0(colnames(gen_dat)[index1], ", (", index1, ")"),
      ylab = paste0(colnames(gen_dat)[index2], ", (", index2, ")"), 
      main = paste(paste0("Pearson of ", round(cor_pearson_mat[index1, index2], 3)),
                   "\n",
                   paste0("Spearman of ", round(cor_spearman_mat[index1, index2], 3))))
}

3-2-1. high pearson and low spearman (linearity vs monotone)

cor_contrast2 <- (abs(cor_pearson_mat) > 0.5) & (abs(cor_spearman_mat) < 0.5)
cor_contrast_ind2 <- which(cor_contrast2, arr.ind = T)
nrow(cor_contrast_ind2)
## [1] 1
  • cannot find one

3-3-1. high pearson and low distance correlation (linearity vs Non-linearity)

cor_contrast3 <- (abs(cor_pearson_mat) > 0.5) & (abs(cor_dist_mat) < 0.5)
cor_contrast_ind3 <- which(cor_contrast3, arr.ind = T)
nrow(cor_contrast_ind3)
## [1] 7

3-3-2. Visualization of high pearson and low distance correlation (linearity vs Non-linearity)

par(mfrow = c(2, 5))

for (i in 1:nrow(cor_contrast_ind3)){
   index1 <- cor_contrast_ind3[i, 1]; index2 <- cor_contrast_ind3[i, 2]
   plot(gen_dat[,index1], gen_dat[,index2], col = gen_label, asp = T,
      pch = 16, xlab = paste0(colnames(gen_dat)[index1], ", (", index1, ")"),
      ylab = paste0(colnames(gen_dat)[index2], ", (", index2, ")"), 
      main = paste(paste0("Pearson of ", round(cor_pearson_mat[index1, index2], 3)),
                   "\n",
                   paste0("Dist.Cor of ", round(cor_dist_mat[index1, index2], 3))))
}

3-4-1. low pearson and high distance correlation(linearity vs Non-linearity)

cor_contrast4 <- (abs(cor_pearson_mat) < 0.5) & (abs(cor_dist_mat) > 0.5)
cor_contrast_ind4 <- which(cor_contrast4, arr.ind = T)
nrow(cor_contrast_ind4)
## [1] 0
  • Cannot find one

3-5-1. low pearson (< 0.5) and high MIC (> 0.5) (linearity vs Information)

cor_contrast5 <- (abs(cor_pearson_mat) < 0.5) & (abs(cor_MIC_mat) > 0.5)
cor_contrast_ind5 <- which(cor_contrast5, arr.ind = T)
nrow(cor_contrast_ind5)
## [1] 4

3-5-2. Visualization of low pearson (< 0.5) and high MIC (> 0.5) (linearity vs Information)

par(mfrow = c(2, 5))

for (i in 1:nrow(cor_contrast_ind5)){
   index1 <- cor_contrast_ind5[i, 1]; index2 <- cor_contrast_ind5[i, 2]
   plot(gen_dat[,index1], gen_dat[,index2], col = gen_label, asp = T,
      pch = 16, xlab = paste0(colnames(gen_dat)[index1], ", (", index1, ")"),
      ylab = paste0(colnames(gen_dat)[index2], ", (", index2, ")"), 
      main = paste(paste0("Pearson of ", round(cor_pearson_mat[index1, index2], 3)),
                   "\n",
                   paste0("MIC of ", round(cor_MIC_mat[index1, index2], 3))))
}

3-6-1. High pearson (> 0.5) and low MIC (< 0.5) (linearity vs Information)

cor_contrast6 <- (abs(cor_pearson_mat) > 0.5) & (abs(cor_MIC_mat) < 0.5)
cor_contrast_ind6 <- which(cor_contrast6, arr.ind = T)
nrow(cor_contrast_ind6)
## [1] 1

3-7-1. Low pearson (< 0.45) and high XI (> 0.55)

cor_contrast7 <- (abs(cor_pearson_mat) < 0.5) & (abs(cor_XI_mat) > 0.5)
cor_contrast_ind7 <- which(cor_contrast7, arr.ind = T)
nrow(cor_contrast_ind7)
## [1] 0

3-8-1. High Pearson and low XI

cor_contrast8 <- (abs(cor_pearson_mat) > 0.5) & (abs(cor_XI_mat) < 0.5)
cor_contrast_ind8 <- which(cor_contrast8, arr.ind = T)
nrow(cor_contrast_ind8)
## [1] 13

3-8-2. Visualization of high Pearson and low XI

par(mfrow = c(2, 5))

for (i in 1:nrow(cor_contrast_ind8)){
   index1 <- cor_contrast_ind8[i, 1]; index2 <- cor_contrast_ind8[i, 2]
   plot(gen_dat[,index1], gen_dat[,index2], col = gen_label, asp = T,
      pch = 16, xlab = paste0(colnames(gen_dat)[index1], ", (", index1, ")"),
      ylab = paste0(colnames(gen_dat)[index2], ", (", index2, ")"), 
      main = paste(paste0("Pearson of ", round(cor_pearson_mat[index1, index2], 3)),
                   "\n",
                   paste0("XI of ", round(cor_XI_mat[index1, index2], 3))))
}

3-9-1. Low Pearson and high Blomqvist’s beta (linearity vs monotone)

cor_contrast9 <- (abs(cor_pearson_mat) < 0.5) & (abs(cor_blomqvist_mat) > 0.5)
cor_contrast_ind9 <- which(cor_contrast9, arr.ind = T)
nrow(cor_contrast_ind9)
## [1] 4
par(mfrow = c(2, 5))
for (i in 1:nrow(cor_contrast_ind9)){
   index1 <- cor_contrast_ind9[i, 1]; index2 <- cor_contrast_ind9[i, 2]
   plot(gen_dat[,index1], gen_dat[,index2], col = gen_label, asp = T,
      pch = 16, xlab = paste0(colnames(gen_dat)[index1], ", (", index1, ")"),
      ylab = paste0(colnames(gen_dat)[index2], ", (", index2, ")"), 
      main = paste(paste0("Pearson of ", round(cor_pearson_mat[index1, index2], 3)),
                   "\n",
                   paste0("Beta of ", round(cor_blomqvist_mat[index1, index2], 3))))
}

3-10-1. High Pearson (> 0.5) and low Blomqvist’s beta (< 0.5) (linearity vs monotone)

cor_contrast10 <- (abs(cor_pearson_mat) > 0.5) & (abs(cor_blomqvist_mat) < 0.5)
cor_contrast_ind10 <- which(cor_contrast10, arr.ind = T)
nrow(cor_contrast_ind10)
## [1] 5

3-10-2. Visualization of High Pearson (> 0.5) and low Blomqvist’s beta (< 0.1) (linearity vs monotone)

par(mfrow = c(2, 5))

for (i in 1:nrow(cor_contrast_ind10)){
   index1 <- cor_contrast_ind10[i, 1]; index2 <- cor_contrast_ind10[i, 2]
   plot(gen_dat[,index1], gen_dat[,index2], col = gen_label, asp = T,
      pch = 16, xlab = paste0(colnames(gen_dat)[index1], ", (", index1, ")"),
      ylab = paste0(colnames(gen_dat)[index2], ", (", index2, ")"), 
      main = paste(paste0("Pearson of ", round(cor_pearson_mat[index1, index2], 3)),
                   "\n",
                   paste0("Beta of ", round(cor_blomqvist_mat[index1, index2], 3))))
}

3-11-1. Low kendall and high Distance Correlation (monotone vs non-linearity)

cor_contrast11 <- (abs(cor_kendall_mat) < 0.5) & (abs(cor_dist_mat) > 0.5)
cor_contrast_ind11 <- which(cor_contrast11, arr.ind = T)
nrow(cor_contrast_ind11)
## [1] 0

3-12-1. high kendall and low Distance Correlation (monotone vs non-linearity)

cor_contrast12 <- (abs(cor_kendall_mat) > 0.5) & (abs(cor_dist_mat) < 0.5)
cor_contrast_ind12 <- which(cor_contrast12, arr.ind = T)
nrow(cor_contrast_ind12)
## [1] 0

3-13-1. Low kendall (< 0.5) and high mutual Information (> 0.5) (monotone vs information)

cor_contrast13 <- (abs(cor_kendall_mat) < 0.5) & (abs(cor_MI_mat) > 0.5)
cor_contrast_ind13 <- which(cor_contrast13, arr.ind = T)
nrow(cor_contrast_ind13)
## [1] 8

3-13-2. Visualization of low kendall (< 0.5) and high mutual Information (> 0.5) (monotone vs information)

par(mfrow = c(2, 5))

for (i in 1:nrow(cor_contrast_ind13)){
   index1 <- cor_contrast_ind13[i, 1]; index2 <- cor_contrast_ind13[i, 2]
   plot(gen_dat[,index1], gen_dat[,index2], col = gen_label, asp = T,
      pch = 16, xlab = paste0(colnames(gen_dat)[index1], ", (", index1, ")"),
      ylab = paste0(colnames(gen_dat)[index2], ", (", index2, ")"), 
      main = paste(paste0("Kendall of ", round(cor_kendall_mat[index1, index2], 3)),
                   "\n",
                   paste0("MI of ", round(cor_MI_mat[index1, index2], 3))))
}

3-14.1. High kendall and low mutual Information (monotone vs information)

cor_contrast14 <- (abs(cor_kendall_mat) > 0.5) & (abs(cor_MI_mat) < 0.5)
cor_contrast_ind14 <- which(cor_contrast14, arr.ind = T)
nrow(cor_contrast_ind14)
## [1] 0

3-15.1. Low Kendall and high XI

cor_contrast15 <- (abs(cor_kendall_mat) < 0.5) & (abs(cor_XI_mat) > 0.5)
cor_contrast_ind15 <- which(cor_contrast15, arr.ind = T)
nrow(cor_contrast_ind15)
## [1] 0

3-16.1. High Kendall and low XI

cor_contrast16 <- (abs(cor_kendall_mat) > 0.5) & (abs(cor_XI_mat) < 0.5)
cor_contrast_ind16 <- which(cor_contrast16, arr.ind = T)
nrow(cor_contrast_ind16)
## [1] 6

3-16.2. Visualization of high Kendall and low XI

par(mfrow = c(2, 5))

for (i in 1:nrow(cor_contrast_ind16)){
   index1 <- cor_contrast_ind16[i, 1]; index2 <- cor_contrast_ind16[i, 2]
   plot(gen_dat[,index1], gen_dat[,index2], col = gen_label, asp = T,
      pch = 16, xlab = paste0(colnames(gen_dat)[index1], ", (", index1, ")"),
      ylab = paste0(colnames(gen_dat)[index2], ", (", index2, ")"), 
      main = paste(paste0("Kendall of ", round(cor_kendall_mat[index1, index2], 3)),
                   "\n",
                   paste0("XI of ", round(cor_XI_mat[index1, index2], 3))))
}

3-17.1. Low distance Correlation and high XI

cor_contrast17 <- (abs(cor_dist_mat) < 0.5) & (abs(cor_XI_mat) > 0.5)
cor_contrast_ind17 <- which(cor_contrast17, arr.ind = T)
nrow(cor_contrast_ind17)
## [1] 0

3-18.1. High distance Correlation and Low XI

cor_contrast18 <- (abs(cor_dist_mat) > 0.5) & (abs(cor_XI_mat) < 0.5)
cor_contrast_ind18 <- which(cor_contrast18, arr.ind = T)
nrow(cor_contrast_ind18)
## [1] 6

3-18.2. Visualization of high distance Correlation and Low XI

par(mfrow = c(2, 5))

for (i in 1:nrow(cor_contrast_ind18)){
   index1 <- cor_contrast_ind18[i, 1]; index2 <- cor_contrast_ind18[i, 2]
   plot(gen_dat[,index1], gen_dat[,index2], col = gen_label, asp = T,
      pch = 16, xlab = paste0(colnames(gen_dat)[index1], ", (", index1, ")"),
      ylab = paste0(colnames(gen_dat)[index2], ", (", index2, ")"), 
      main = paste(paste0("Dist.Cor of ", round(cor_dist_mat[index1, index2], 3)),
                   "\n",
                   paste0("XI of ", round(cor_XI_mat[index1, index2], 3))))
}

3-19.1. Low Blomqvist’s Beta and high XI

cor_contrast19 <- (abs(cor_blomqvist_mat) < 0.5) & (abs(cor_XI_mat) > 0.5)
cor_contrast_ind19 <- which(cor_contrast19, arr.ind = T)
nrow(cor_contrast_ind19)
## [1] 0

3-20.1. High Blomqvist’s Beta (> 0.65) and low XI (< 0.45)

cor_contrast20 <- (abs(cor_blomqvist_mat) > 0.65) & (abs(cor_XI_mat) < 0.45)
cor_contrast_ind20 <- which(cor_contrast20, arr.ind = T)
nrow(cor_contrast_ind20)
## [1] 8

3-20.2. Visualization of High Blomqvist’s Beta (> 0.65) and low XI (< 0.45)

par(mfrow = c(2, 5))

for (i in 1:nrow(cor_contrast_ind20)){
   index1 <- cor_contrast_ind20[i, 1]; index2 <- cor_contrast_ind20[i, 2]
   plot(gen_dat[,index1], gen_dat[,index2], col = gen_label, asp = T,
      pch = 16, xlab = paste0(colnames(gen_dat)[index1], ", (", index1, ")"),
      ylab = paste0(colnames(gen_dat)[index2], ", (", index2, ")"), 
      main = paste(paste0("Beta of ", round(cor_blomqvist_mat[index1, index2], 3)),
                   "\n",
                   paste0("XI of ", round(cor_XI_mat[index1, index2], 3))))
}